UNIT - II TIME SERIESAccording to Ya-Lun-Chou, “A time series may be deﬁ ned as a collection of reading belonging to different time period of same economic variable or composite

CHAPTER OVERVIEW

Time Series

Measures of Secular

Trend

Simple Average Method

Graphic or a free

hand curve method

Ratio to Trend

Method

Method of Semi

-averages

Link Relatives Method

Method of Moving Averages

Cyclical Variations

Method of Least Squares

Random or Irregular

Variations

AdditiveModel

Multiplicative Model

Components of Time Series Models of Time Series

At the end of this Chapter, you will be able to:

w Understand the components of Time Series

w Calculate the trend using graph, moving averages

w Calculate Seasonal variations for both Additive and Multiplicative models

LEARNING OBJECTIVES

Secular Trend

Seasonal Variations

UNIT - II TIME SERIES

JSNR_51703829_ICAI_Business Mathematics_Logical Reasoning & Statistice_Text.pdf___772 / 808© The Institute of Chartered Accountants of India

19.39TIME SERIES

19.2.1 INTRODUCTIONWe came across a data which is collected on a variable/s (rainfall, production of industrial product, production of rice, sugar cane, import/export of a country, population, etc.) at different time epochs (hours, days, months, years etc.), such a data is called time series data. Time series is statistical data that are arranged and presented in a chronological order i.e., over a period of time.

Most of the time series relating to Economic, Business and Commerce might show an upward tendency in case of population, production & sales of products, incomes, prices; or downward tendency might be noticed in time series relating to share prices, death, birth rate etc. due to global melt down, or improvement in medical facilities etc.

Defi nition: According to Spiegel, “A time series is a set of observations taken at specifi ed times, usually at equal intervals.”

According to Ya-Lun-Chou, “A time series may be defi ned as a collection of reading belonging to different time period of same economic variable or composite of variables.”

Components of Time Series: There are various forces that affect the values of a phenomenon in a time series; these may be broadly divided into the following four categories, commonly known as the components of a time series.

(1) Long term movement or Secular Trend

(2) Seasonal variations

(3) Cyclical variations

(4) Random or irregular variations

In traditional or classical time series analysis, it is ordinarily assumed that there are:

1. Secular Trend or Simple trend:

Secular trend is the long: Term tendency of the time series to move in an upward or down ward direction. It indicates how on the whole, it has behaved over the entire period under reference. These are result of long-term forces that gradually operate on the time series variable. A general tendency of a variable to increase, decrease or remain constant in long term (though in a small time interval it may increase or decrease) is called trend of a variable. E.g. Population of a country has increasing trend over a years. Due to modern technology, agricultural and industrial production is increasing. Due to modern technology health facilities, death rate is decreasing and life expectancy is increasing. Secular trend is be long-term tendency of the time series to move on upward or downward direction. It indicates how on the whole behaved over the entire period under reference. These are result of long term forces that gradually operate on the time series variable. A few examples of theses long term forces which make a time series to move in any direction over long period of the time are long term changes per capita income, technological improvements of growth of population, Changes in Social norms etc.

Most of the time series relating to Economic, Business and Commerce might show an upward tendency in case of population, production & sales of products, incomes, prices; or downward tendency might be noticed in time series relating to share prices, death, birth rate etc. due to global melt down, or improvement in medical facilities etc. All these indicate trend.


19.40 STATISTICS

2. Seasonal variations:

Over a span of one year, seasonal variation takes place due to the rhythmic forces which operate in a regular and periodic manner. These forces have the same or almost similar pattern year after year.

It is common knowledge that the value of many variables depends in part on the time of year. For Example, Seasonal variations could be seen and calculated if the data are recorded quarterly, monthly, weekly, daily or hourly basis. So if in a time series data only annual fi gures are given, there will be no seasonal variations.

The seasonal variations may be due to various seasons or weather conditions for example sale of cold drink would go up in summers & go down in winters. These variations may be also due to man-made conventions & due to habits, customs or traditions. For example, sales might go up during Diwali & Christmas or sales of restaurants & eateries might go down during Navratri’s.

The method of seasonal variations are

(i) Simple Average Method

(ii) Ratio to Trend Method

(iii) Ratio to Moving Average Method

(iv) Link Relatives Method

3. Cyclical variations:

Cyclical variations, which are also generally termed as business cycles, are the periodic movements. These variations in a time series are due to ups & downs recurring after a period from Season to Season. Though they are more or less regular, they may not be uniformly periodic. These are oscillatory movements which are present in any business activity and is termed as business cycle. It has got four phases consisting of prosperity (boom), recession, depression and recovery. All these phases together may last from 7 to 9 years may be less or more.

4. Random or irregular variations:

These are irregular variations which occur on account of random external events. These variations either go very deep downward or too high upward to attain peaks abruptly. These fl uctuations are a result of unforeseen and unpredictably forces which operate in absolutely random or erratic manner. They do not have any defi nite pattern and it cannot be predicted in advance. These variations are due to fl oods, wars, famines, earthquakes, strikes, lockouts, epidemics etc.

19.2.2 MODELS OF TIME SERIESThe following are the two models which are generally used for decomposition of time series into its four components. The objective is to estimate and separate the four types of variations and to bring out the relative impact of each on the overall behaviour of the time series.

(1) Additive model

(2) Multiplicative model

Additive Model: In additive model it is assumed that the four components are independent of one another i.e. the pattern of occurrence and magnitude of movements in any particular component does not affect and are not affected by the other component. Under this assumption the four components are arithmetically additive ie. magnitude of time series is the sum of the separate infl uences of its four components i.e. Yt= T + C + S + I


19.41TIME SERIES

Where,

Yt = Time series

T = Trend variation

C = Cyclical variation

S = Seasonal variation

I = Random or irregular variation

Multiplicative Model: In this model it is assumed that the forces that give rise to four types of variations are interdependent, so that overall pattern of variations in the time series is a combined result of the interaction of all the forces operating on the time series. Accordingly, time series are the product of its four components i.e.

Yt = T x C x S x I

As regards to the choice between the two models, it is generally the multiplication model which is used more frequently. As the forces responsible for one type of variation are also responsible for other type of variations, hence it is multiplication model which is more suited in most business and economic time series data for the purpose of decomposition.

Example 19.2.1: Under the additive model, a monthly sale of ` 21,110 explained as follows:

The trend might be ` 20,000, the seasonal factor: ` 1,500 (The month question is a good one for sales, expected to be ` 1,500 over the trend), the cyclical factor: ` 800 (A general Business slump is experienced, expected to depress sales by ` 800 (per month); and Residual Factor: ` 410 (Due to unpredictable random fl uctuations).

The model gives:

Y = T + C + S + R

21,110 = 2,000 + 1,500 + (-800) + 410

The multiplication model might explain the same sale fi gure in similar way.

Trend = ` 20,000, Seasonal Factor: ` 1.15 (a good month for sales, expected to be 15 per cent above the trend)

Cyclical Factor: 0.90 (a business slump, expected to cause a 10 per cent reduction in sales) and

Residual Factor: 1.02 (Random fl uctuations of + 2 Factor)

Y = T × S × C × R

21114 = 20,000 × 1.15 × 0.90 × 1.02

19.2.3 MEASUREMENT OF SECULAR TRENDThe following are the methods most commonly used for studying & measuring the trend component in a time series -

(1) Graphic or a Freehand Curve Method

(2) Method of Semi Averages

(3) Method of Moving Averages

(4) Method of Least Squares


19.42 STATISTICS

Graphic or Freehand Curve Method:

The data of a given time series is plotted on a graph and all the points are joined together with a straight line. This curve would be irregular as it includes short run oscillation. These irregularities are smoothened out by drawing a freehand curve or line along with the curve previously drawn.

This curve would eliminate the short run oscillations and would show the long period general tendency of the data. While drawing this curve it should be kept in mind that the curve should be smooth and the number of points above the trend curve should be more or less equal to the number of points below it.

Merits:

(1) It is very simple and easy to understand

(2) It does not require any mathematical calculations

Disadvantages:

(1) This is a subjective concept. Hence different persons may draw freehand lines at different positions and with different slopes.

(2) If the length of period for which the curve is drawn is very small, it might give totally erroneous results.

Example 19.2.2: The following are fi gures of a Sale for the last nine years. Determine the trend by line by the freehand method.

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008

Sale in lac Units 75 95 115 65 120 100 150 135 175

7595

115

65

120100

150 135

175

0

50

100

150

200

2000 2001 2002 2003 2004 2005 2006 2007 2008

Sale

in L

ac U

nits

Years

Sale in Lac Units

Sale in Lac Units Linear (Sale in Lac Units )

The trend line drawn by the freehand method can be extended to predicted values. However, since the freehand curve fi tting is too subjective, the method should not be used as basis for predictions.

Methods of Semi averages: Under this method the whole time series data is classifi ed into two equal parts and the averages for each half are calculated. If the data is for even number of years, it is easily divided into two. If the data is for odd number of years, then the middle year of the time series is left and the two halves are constituted with the period on each side of the middle year.

The arithmetic mean for a half is taken to be representative of the value corresponding to the midpoint of the time interval of that half. Thus we get two points. These two points are plotted on a graph and then are joined by straight line which is our required trend line.


19.43TIME SERIES

Example 19.2.3: Fit a trend line to the following data by the method of Semi-averages.

Year 2000 2001 2002 2003 2004 2005 2006

Sale in lac Units 100 105 115 110 120 105 115

Solution: Since the data consist of seven Years, the middle year shall be left out and an average of the

fi rst three years and last three shall be obtained. The average of fi rst three years is 100 105 115

3

or 320

3 or 106.67 and the average of last three years 120 105 115

3

or 340

3 or 133.33.

100

105

115

110

120

105

115

90

95

100

105

110

115

120

125

2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6

SALE

S IN

TH

OU

SAN

D U

NIT

S

YEAR

Sales of a Firm A (Thousands Units)

Moving average method: A moving average is an average (Arithmetic mean) of fi xed number of items (known as periods) which moves through a series by dropping the fi rst item of the previously averaged group and adding the next item in each successive average. The value so computed is considered the trend value for the unit of time falling at the centre of the period used in the calculation of the average.

In case the period is odd: If the period of moving average is odd for instance for computing 3 yearly moving average, the value of 1st, 2nd and 3rd years are added up and arithmetic mean is found out and the answer is placed against the 2nd year; then value of 2nd, 3rd and 4th years are added up and arithmetic mean is derived and this average is placed against 3rd year (i.e. the middle of 2nd, 3rd and 4th) and so on.

In case of even number of years: If the period of moving average is even for instance for computing 4 yearly moving average, the value of 1st, 2nd, 3rd and 4th years are added up& arithmetic mean is found out and answer is placed against the middle of 2nd and 3rd year. The second average is placed against middle of 3rd & 4th year. As this would not coincide with a period of a given time series an attempt is made to synchronise them with the original data by taking a two period average of the moving averages and placing them in between the corresponding time periods. This technique is called centring& the corresponding moving averages are called moving average centred.

Example 19.2.4: The wages of certain factory workers are given as below. Using 3 yearly moving average indicate the trend in wages.

Year 2004 2005 2006 2007 2008 2009 2010 2011 2012

Wages 1200 1500 1400 1750 1800 1700 1600 1500 1750


19.44 STATISTICS

Solution:

Table: Calculation of Trend Values by method of 3 yearly Moving Average

Year Wages 3 yearly movingtotals

3 yearly movingaverage i.e. trend

2004 1200 – –

2005 1500 4100 1366.67

2006 1400 4650 1550

2007 1750 4950 1650

2008 1800 5250 1750

2009 1700 5100 1700

2010 1600 4800 1600

2011 1500 4850 1616.67

2012 1750 – –

Example 19.2.5: Calculate 4 yearly moving average of the following data.

Year 2005 2006 2007 2008 2009 2010 2011 2012

Wages 1150 1250 1320 1400 1300 1320 1500 1700

Solution: (First Method):

Table: Calculation of 4 year Centred Moving Average

Year Wages 4 yearly moving 2 year moving total 4 yearly moving

(1) (2) total (3) of col. 3 (centred) (4) average centred (5) [col. 4/8]

2005 1,150 – – –2006 1,250 – – –

5,1202007 1,320 10,390 1,298.75

5,2702008 1,400 10,610 1,326.25

5,3402009 1,300 10,860 1,357.50

5,5202010 1,320 11,340 1,417.50

5,8202011 1,5002012 1,700


19.45TIME SERIES

Second Method:

Table: Calculation of 4 year Centred Moving Average

Year Wages 4 yearly moving total (3)

4 yearly moving average (4)

2 year moving total of col. 4 (centered) (5)

4 year centered moving average

(col. 5/2)2005 1,150 – – – –2006 1,250 – – – – –

5,120 1,280 – –2007 1,320 2,597.75 1,298.75

5,270 1,317.5 – –2008 1,400 2,652.5 1,326.25

5,340 1,335 – –2009 1,300 2,715 1,357.50

5,520 1,3802010 1,320 2,835 1,417.50

5,820 1,455 – –2011 1,5002012 1,700

Example 19.2.6: Calculate fi ve yearly moving averages for the following data.

Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Value 123 140 110 98 104 133 95 105 150 135

Solution:

Table: Computation of Five Yearly Moving Averages

Year Value (‘000 `)

5 yearly moving totals (‘000 `)

5 yearly moving average(‘000 `)

2003 123 – –2004 140 – –2005 110 575 1152006 98 585 1172007 104 540 1082008 133 535 1072009 95 587 117.42010 105 618 123.62011 150 – –2012 135 – –

The method of least squares as studied in regression analysis can be used to fi nd the trend line of best fi t to a time series data.

The regression trend line (Y) is defi ned by the following equation – Y = a + bX


19.46 STATISTICS

where Y = predicted value of the dependent variable

a = Y axis intercept or the height of the line above origin (i.e. when X = 0, Y = a)

b = slope of the regression line (it gives the rate of change in Y for a given change in X) (when b is positive the slope is upwards, when b is negative, the slope is downwards) X = independent variable (which is time in this case)

To estimate the constants a and b, the following two equations have to be solved simultaneously –

∑Y = na + b∑X

∑XY = a∑X + b∑X2

To simplify the calculations, if the mid point of the time series is taken as origin, then the negative values in the fi rst half of the series balance out the positive values in the second half so that ∑x = 0. In this case the above two normal equations will be as follows -

∑Y = na

∑XY = b∑X2

In such a case the values of a and b can be calculated as under -

Since ∑Y = na,

Since ∑XY = b∑X2

b = 2

XYX

; a = Yn

Example 19.2.7: Fit a straight line trend to the following data by Least Square Method and estimate the sale for the year 2012.

Year 2005 2006 2007 2008 2009 2010

Sale (in’000s) 70 80 96 100 95 114

Solution:

Table: Calculation of trend line

Year SalesY

Deviations from 2007.5

Deviationsmultiplied by 2 (X)

X2 XY

2005 70 -2.5 -5 25 -350

2006 80 -1.5 -3 9 -240

2007 96 -.5 -1 1 -96

2008 100 +.5 +1 1 100

2009 95 +1.5 +3 9 285

2010 114 +2.5 +5 25 570

∑Y = 555 ∑X2 = 70 ∑XY = 269


19.47TIME SERIES

N = 6

Equation of the straight line trend is Yo = a + bX

a= 55592.5

6

Y=

N ; b = 2

2693.843

70

XY

X

Trend equation is Yc = 92.5 + 3.843X

For 2012, X = 9

Y2012 = 92.5 + 3.843 × 9 = 92.5 + 34.587

= 126.59 (in ‘000 `)

Example 19.2.8:

Fit a straight line trend to the following data and estimate the likely profi t for the year 2012. Also calculate the trend values.

Year 2003 2004 2005 2006 2007 2008 2009

Profi t (in lakhs of `) 60 72 75 65 80 85 95

Solution:

Table: Calculation of Trend and Trend Values

Year Profi tY

Deviationfrom 2006

X2 XY Trend Values(Yc = a + bX)

X [Yc= 76 + 4.85X]2003 60 -3 9 -180 76 + 4.85 (-3) = 61.452004 70 -2 4 -144 76 + 4.85 (-2) = 66.302005 75 -1 1 -75 76 + 4.85 (-1) = 70.152006 65 0 0 0 76 + 4.85 (0) = 762007 80 1 1 80 76 + 4.85(1) = 80.852008 85 2 4 170 76 + 4.85 (2) = 85.702009 95 3 9 285 76 + 4.85 (3) = 90.55

∑y = 532 ∑X2 = 28 ∑XY=136

N = 7

The equation for straight line trend is Yc = a + bX

Where

a = 55592.5

6

Y

N

b= 2

2693.843

70

XY

X


19.48 STATISTICS

The trend equation Yc = 92.5 + 3.843.X

2012, x = 6 (2012 - 2006) Yc = 76 + 4.85(6) = 76 + 29.10

= 105.10

The estimated profi t for the year 2012 is ` 105.10 lakhs.

Example 19.2.9: Calculate Seasonal Indices for each quarter from the following percentages of whole sale price indices to their moving averages.

Year Quarter

I II III IV

2003 - - 11.0 11.02004 12.5 13.5 15.5 14.52005 16.8 15.2 13.1 15.32006 11.2 11.0 12.4 13.22007 10.5 13.3 -

Solution:

Year Quarter

I II III IV

Quarterly Total 51.0 53.0 52.0 54.0Quarterly Average 12.75 13.25 13.0 13.5

Average of the Quarterly Averages = 52.5 13.1254

Year Quarter

I II III IV

Seasonal Indices 12.75 100

13.12597.143

13.25 100

13.125100.952

13.0 100

13.12597.143

13.5 100

13.125102.857

Seasonal Indices are calculated by converting the respective quarterly averages on the basis that the average of the quarterly average = 100

Example 19.2.10: Calculate 5- year weighted moving averages for the following data, using weights 1, 1, 3, 2 respectively:

Year 1 2 3 4 5 6 7 8 9 10

Codded Sales 40 33 72 81 76 68 91 87 98 97


19.49TIME SERIES

Solution :

Year I Sales II 5- Year Weighted Average IV1 40 – –2 33 – –

3 72 140 1 33 1 72 3 81 2 76 1

8 = 65.8

4 81 133 1 72 1 81 3 76 2 68 1

8 = 71.00

5 76 172 1 81 1 76 3 68 2 91 1

8 = 76.00

6 68 181 1 76 1 68 3 91 2 87 1

8 = 78.750

7 91 176 1 68 1 91 3 87 2 90 1

8 = 86.125

8 87 168 1 91 1 87 3 98 2 97 1

8 = 89.125

9 98 – –10 97 – –

Example 19.2.11: Assuming no trend, calculate Seasonal variation indices for the following data.

Year Quarterly DataQ1 Q2 Q3 Q4

2013 3.7 4.1 3.3 3.52014 3.7 3.9 3.6 3.62015 4.0 4.1 3.3 3.12016 3.3 4.4 4.0 4.0

Solution:

Quarterly DataQ1 Q2 Q3 Q43.7 4.1 3.3 3.53.7 3.9 3.6 3.64.0 4.1 3.3 3.13.3 4.4 4.0 4.0

Quarterly Total 14.7 16.5 14.2 14.2Quarterly Average 3.675 4.125 3.55 3.55


19.50 STATISTICS

Average of Quarterly Averages = 3.675 4.125 3.55 3.55 14.93.725

4 4

Example 19.2.12: Calculate the Seasonal Indices from the following ratio to moving averages values expressed in percentage.

YearSeasons

Summer Rain Winter2009 – 101.75 107.142010 96.18 92.30 114.002011 92.45 95.20 118.18

Solution:

YearSeasons

Summer Rain Winter2009 – 101.75 107.142010 96.18 92.30 114.002011 92.45. 95.20 118.18Total 188.63 289.25 339.832Average 94.315 96.417 113.107 303.839Constructed Seasonal Index 93.127 95.202 111.682

Correction Factor = 3000.9874

303.839

Example 19.2.13: From the following data, calculate the trend values, using Four yearly moving average.

Years 2008 2009 2010 2011 2012 2013 2014 2015 2016Values 506 620 1036 673 588 696 1116 738 663

Solution:

Year Values 4 Yearly Moving Totals (a)

2 Period Moving Total of (a)

4 Yearly Moving Averages

2008 506 – – –2009 620 – – –2010 1036 2835 5752 719.02011 673 2917 5910 738.82012 588 2993 6066 758.32013 696 3073 6211 776.42014 1116 3138 6311 793.92015 738 3213 – –2016 663 – – –


19.51TIME SERIES

1) A time series is set of measurements on a variable taken over some period of time, it has four components.

(a) Trend (b) Seasonal variations

(c) Cyclical variations (d) Irregular variations

2) There are two models of time series

(a) Additive Model (b) Multiplicative Model

3) Trends can be measured in the following measures

(a) Free hand curve method (b) Semi-averages method

(c) Moving averages method (d) Least squares method

4) A time series may be determined by eliminating the computed trend values from the given data set. It may done using additive model or multiplicative model.

5) Seasonal variations can be measured in any of the following methods:

(a) simple averages (b) Ratio to trend method

(c) Ratio to Moving averages (d) Link relative method

6) Time series data can be deseaonalised by eliminating the effect of seasonal variations from it.

7) Irregular component in a time series is measured as a residue after eliminating all other fl uctuations from data.

8) Time Series is useful in forecasting future values.

Unit -II Exercise 1 (a)

1) (a) What is meant by Time Series? Explain its utility.

(b) Explain clearly the meaning of Time Series Analysis in business.

2) What is a Time Series? What are the different components? Describe briefl y each of them.

3) Write Short Notes on:

(a) Moving average method of measuring Trend

(b) Seasonal Index

4) Calculate fi ve yearly moving averages for the following data.

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Value (‘000 `)

123 140 110 98 104 133 95 105 150 135

5) Write Short notes on:

(i) Seasonal variations and (ii) Cyclical variations

SUMMARY


19.52 STATISTICS

6) From the following data verify that 5 year weighted moving average with weights 1, 2, 2, 2, 1 respectively is equivalent to the 4 year centred moving average:

(` In lakhs)Year 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009Sale 5 3 7 6 4 8 9 10 8 9 9

7) Explain the additive and multiplicative models of Time Series.

8) Calculate the Seasonal Indices by the method of Link Relatives for the following data.

Quarter Quarterly Figures for Five years

2003 2004 2005 2006 2007

I 45 48 49 52 60

II 54 56 63 65 70

III 72 63 70 75 83

IV 60 56 65 72 86

9) Calculate the seasonal Indices for each quarter from the following percentages of whole sale prices indices to their moving averages.

Year QuarterI II III IV

2003 – – 11.0 802004 12.5 13.5 15.5 14.52005 16.8 15.2 13.1 15.32006 11.2 11.0 12.4 13.22007 10.5 13.3 – –

10) Assuming no trend in the series, Calculate seasonal indices for the following data.

Year QuarterI II III IV

2004 78 66 84 802005 76 74 82 782006 72 68 80 702007 74 70 84 742008 76 74 86 82

11) The annual production in commodity is given as follows:

Year 2000 2001 2002 2003 2004 2005 2006Production:(in tonnes) 70 80 90 95 102 110 115

(a) Fit a straight line trend by the method of least squares.

(b) Convert the annual trend equation into monthly trend equation.


19.53TIME SERIES

Unit -II Exercise 1(B)

Choose the most appropriate option (a) or (b) or (c) or (d).

1) An orderly set of data arranged in accordance with their time of occurrence is called:

(a) Arithmetic series (b) Harmonic series

(c) Geometric series (d) Time series

2) A time series consists of:

(a) Short-term variations (b) Long-term variations

(c) Irregular variations (d) All of the above

3) The graph of time series is called:

(a) Histogram (b) Straight line

(c) Histogram (d) Ogive

4) Secular trend can be measured by:

(a) Two methods (b) Three methods

(c) Four methods (d) Five methods

5) The secular trend is measured by the method of semi-averages when:

(a) Time series based on yearly values

(b) Trend is linear

(c) Time series consists of even number of values

(d) None of them

6) Increase in the number of patients in the hospital due to heat stroke is:

(a) Secular trend (b) Irregular variation

(c) Seasonal variation (d) Cyclical variation

7) The systematic components of time series which follow regular pattern of variations are called:

(a) Signal (b) Noise

(c) Additive model (d) Multiplicative model

8) The unsystematic sequence which follows irregular pattern of variations is called:

(a) Noise (b) Signal

(c) Linear (d) Non-linear

9) In time series seasonal variations can occur within a period of:

(a) Four years (b) Three years

(c) One year (d) Nine years

10) Wheat crops badly damaged on account of rains is:

(a) Cyclical movement (b) Random movement


19.54 STATISTICS

(c) Secular trend (d) Seasonal movement

11) The method of moving average is used to fi nd the:

(a) Secular trend (b) Seasonal variation

(c) Cyclical variation (d) Irregular variation

12) Most frequency used mathematical model of a time series is:

(a) Additive model (b) Mixed model

(c) Multiplicative model (d) Regression

13) A time series consists of:

(a) No mathematical model (b) One mathematical model

(c) Two mathematical models (d) Three mathematical models

14) In semi-averages method, we decide the data into:

(a) Two parts (b) Two equal parts

(c) Three parts (d) Diffi cult to tell

15) Moving average method is used for measurement of trend when:

(a) Trend is linear (b) Trend is non-linear

(c) Trend is curvi linear (d) None of them

16) When the trend is of exponential type, the moving averages are to be computed by using:

(a) Arithmetic mean (b) Geometric mean

(c) Harmonic mean (d) Weighted mean

17) The long term trend of a time series graph appears to be:

(a) Straight-line (b) Upward

(c) Downward (d) Parabolic curve or third degree curve

18) Indicate which of the following an example of seasonal variations is:

(a) Death rate decreased due to advance in science

(b) The sale of air condition increases during summer

(c) Recovery in business

(d) Sudden causes by wars

19) The most commonly used mathematical method for measuring the trend is:

(a) Moving average method (b) Semi average method

(c) Method of least squares (d) None of them

20) A trend is the better fi tted trend for which the sum of squares of residuals is:

(a) Maximum (b) Minimum

(c) Positive (d) Negative


19.55TIME SERIES

21) Decomposition of time series is called:

(a) Historigram (b) Analysis of time series

(c) Histogram (d) Detrending

22) The fi re in a factory is an example of:

(a) Secular trend (b) Seasonal movements

(c) Cyclical variations (d) Irregular variations

23) Increased demand of admission in the subject of computer in Uttar Pradesh is:

(a) Secular trend (b) Cyclical trend

(c) Seasonal trend (d) Irregular trend

24) Damages due to fl oods, droughts, strikes fi res and political disturbances are:

(a) Trend (b) Seasonal

(c) Cyclical (d) Irregular

25) The general pattern of increase or decrease in economics or social phenomena is shown by:

(a) Seasonal trend (b) Cyclical trend

(c) Secular trend (d) Irregular trend

26) In moving average method, we cannot fi nd the trend values of some:

(a) Middle periods (b) End periods

(c) Starting periods (d) Between extreme periods

27) Moving-averages:

(a) Give the trend in a straight line (b) Measure the seasonal variations

(c) Smooth-out the time series (d) None of them

28) The rise and fall of a time series over periods longer than one year is called:

(a) Secular trend (b) Seasonal variation

(c) Cyclical variation (d) Irregular variations

29) A time series has:

(a) Two Components (b) Three Components

(c) Four Components (d) Five Components

30) The multiplicative time series model is:

(a) Y = T + S + C + I (b) Y = TSCI

(c) Y= a + bx (d) y = a + bx + C x 2

31) The additive model of Time Series

(a) Y = T + S + C + I (b) Y = TSCI

(c) Y= a + bx (d) y = a + bx + C x 2


19.56 STATISTICS

32) A pattern that is repeated throughout a time series and has a recurrence period of at most one year is called:

(a) Cyclical variation (b) Irregular variation

(c) Seasonal variation (d) Long term variation

33) If an annual time series consisting of even number of years is coded, then each coded interval is equal to:

(a) Half year (b) One year

(c) Both (a) and (b) (d) Two years

34) In semi averages method, if the number of values is odd then we drop:

(a) First value (b) Last value

(c) Middle value (d) Middle two values

35) The trend values in freehand curve method are obtained by:

(a) Equation of straight line (b) Graph

(c) Second degree parabola (d) All of the above

ANSWERS

Unit-II

Exercise 1(a)

4) 115, 117, 108, 107, 117.4, 123.6

6) 5.125, 5.625, 6.500, 7.250, 5.475, 5.700

8) 82.86, 98.45, 114.60, 104.08

11) a) y = 94.6 + 7.39 x (b) Monthly trend equation is y = 7.88 + 0.05 x

Exercise 1(b)

1. (d) 2. ( d) 3. (c) 4. (c ) 5. (b) 6. (c) 7. (a) 8. (a) 9. (c) 10. (b)

11. (a) 12. (c) 13. (d) 14. (b) 15. (a) 16. (d) 17. (b) 18. (c) 19. (b) 20. (b)

21. (d) 22. (a) 23. (d) 24. (c) 25. (d) 26. (c) 27. (c) 28. (c) 29. (c) 30. (b)

31. (b) 32. (c) 33. (c) 34. (c) 35. (b)


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UNIT - II TIME SERIESAccording to Ya-Lun-Chou, “A time series may be deﬁ ned as a collection of reading belonging to different time period of same economic variable or composite